Arithmetic Sequence Examples:
For example, consider 4, 7,10.... In this sequence, first term is 4 and we get the second term as 7 = 4 + 3. Second term is 7 and we get third term as 10 = 7 + 3. So, it is clear that the difference between the two successive terms is 3 and it is the same for all terms or in the complete sequence. So, this is the example for arithmetic sequence.
Consider the sequence 2, 7, 12, 17, 22,.......
1. The nth term of any arithmetic sequence is given by Tn = a + (n - 1)d
2. The 6th term(taking n = 6), T6 = 2 + ( 6 -1) 5 = 27
3. The 8th term(taking n = 8), T8 = 2 + ( 8 -1) 5 = 37
In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern. But the sequence is generated by the recurrence relation and is given by
a1 = a2 = 1
a3 = a1 + a2
an = an - 2 + an - 1, n > 2
If we have a sequence as x1, x2 ,x3,......., and the sequence follows the rule as xj - xi = d for all values of i and j, then the sequence is called the arithmetic sequence. This type of sequence where any term except the first term is obtained by adding a fixed number to the previous term.
Arithmetic Sequence Formula:
If we have x1, x2 , x
n,..... be the any arithmetic sequence, then the sum of this finite series, where n is any positive integer is given by x1+x2+..............+xn=∑ni=1xi
Geometric SequenceA sequence where any term except the first term is obtained by multiplying a fixed number to the previous term is called geometric sequence. If the sequence x1, x2, x3,.......,xn follows the condition xn+1xnxn+1xn = r, then that sequence is known as geometric sequence and r is known as the common ration of the sequence.
Geometric Sequence Examples:
Consider the sequence 10, 20, 40, 80
Here, a = 10, r = 2
5th term is T5 = 10 x (2) 5 - 1 = 10 x (2) 4 = 160
7th term is T7 = 10 x (2) 7 - 1 = 10 x (2) 6 = 640
Harmonic SequenceThis is a type of sequence where the reciprocal of the terms from an arithmetic sequence. So, if any sequence is in the form of 1x1,1x2,1x3,.......,1xn1x1,1x2,1x3,.......,1xn, then the sequence is said to be harmonic sequence.
If a, b and c are in the form of arithmetic sequence, then b = 2aca+c2aca+cFor example, the sequence 11,12,13,.....11,12,13,..... is a harmonic sequence.
(Sorry nga pa la.. wag ka sana magalit)